# Motion With Constant Acceleration (Examples, Exercises, Solutions)

An abstraction often used in physics is motion with constant acceleration. This is a good approximation for many different situations: free fall over small distances or in low-density atmospheres, full braking in car traffic, an object sliding down an inclined plane, etc … The mathematics behind this special case is relatively simple. Assume the object that is subject to the constant acceleration a (in m/s²) initially has a velocity v(0) (in m/s). Since the velocity is the integral of the acceleration function, the object’s velocity after time t (in s) is simply:

1) v(t) = v(0) + a · t

For example, if a car initially goes v(0) = 20 m/s and brakes with a constant a = -10 m/s², which is a realistic value for asphalt, its velocity after a time t is:

v(t) = 20 – 10 · t

After t = 1 second, the car’s speed has decreased to v(1) = 20 – 10 · 1 = 10 m/s and after t = 2 seconds the car has come to a halt: v(2) = 20 – 10 · 2 = 0 m/s. As you can see, it’s all pretty straight-forward. Note that the negative acceleration (also called deceleration) has led the velocity to decrease over time. In a similar manner, a positive acceleration will cause the speed to go up. You can read more on acceleration in this blog post.

What about the distance x (in m) the object covers? We have to integrate the velocity function to find the appropriate formula. The covered distance after time t is:

2) x(t) = v(0) · t + 0.5 · a · t²

While that looks a lot more complicated, it is really just as straight-forward. Let’s go back to the car that initially has a speed of v(0) = 20 m/s and brakes with a constant a = -10 m/s². In this case the above formula becomes:

x(t) = 20 · t – 0.5 · 10 · t²

After t = 1 second, the car has traveled x(1) = 20 · 1 – 0.5 · 10 · 1² = 15 meters. By the time it comes to a halt at t = 2 seconds, it moved x(2) = 20 · 2 – 0.5 · 10 · 2² = 20 meters. Note that we don’t have to use the time as a variable. There’s a way to eliminate it. We could solve equation 1) for t and insert the resulting expression into equation 2). This leads to a formula connecting the velocity v and distance x.

3)

Solved for x it looks like this:

3)’

It’s a very useful formula that you should keep in mind. Suppose a tram accelerates at a constant a = 1.3 m/s², which is also a realistic value, from rest (v(0) = 0 m/s). What distance does it need to go to full speed v = 10 m/s? Using equation 3)’ we can easily calculate this:

————————————————————————————-

Here are a few exercises and solutions using the equations 1), 2) and 3).

1. During free fall (air resistance neglected) an object accelerates with about a = 10 m/s. Suppose the object is dropped, that is, it is initially at rest (v(0) = 0 m/s).

a) What is its speed after t = 3 seconds?
b) What distance has it traveled after t = 3 seconds?
c) Suppose we drop the object from a tower that is x = 20 meters tall. At what speed will it impact the ground?
d) How long does the drop take?

Hint: in exercise d) solve equation 1) for t and insert the result from c)

2. During the reentry of space crafts accelerations can be as high as a = -70 m/s². Suppose the space craft initially moves with v(0) = 6000 m/s.

a) What’s the speed and covered distance after t = 10 seconds?
b) How long will it take the space craft to half its initial velocity?
c) What distance will it travel during this time?

3. An investigator arrives at the scene of a car crash. From the skid marks he deduces that it took the car a distance x = 55 meters to come to a halt. Assume full braking (a = -10 m/s²). Was the car initially above the speed limit of 30 m/s?

————————————————————————————-

Solutions to the exercises:

Exercise 1

a) 30 m/s
b) 45 m
c) 20 m/s
d) 2 s

Exercise 2

a) 5,300 m/s and 56,500 m
b) 42.9 s (rounded)
c) 192,860 m (rounded)

Exercise 3

Yes (he was initially going 33.2 m/s)

————————————————————————————-

To learn the basic math you need to succeed in physics, check out the e-book “Algebra – The Very Basics”. For an informal introduction to physics, check out the e-book “Physics! In Quantities and Examples”. Both are available at low prices and exclusively for Kindle.

# The Doppler Effect in Pictures

The siren of an approaching police car will sound at a higher pitch, the light of an approaching star will be shifted towards blue and a passing supersonic jet will create a violent thunder. What do these phenomenon have in common? All of them are a result of the Doppler effect. To understand how it arises, just take a look at the animations below.

Stationary Source: The waves coming from the source propagate symmetrically.

Subsonic Source (moving below sound speed): Compression of waves in direction of motion.

Sonic Source (moving at sound speed): Maximum compression.

Supersonic Source (moving beyond sound speed): Source overtakes its waves, formation of Mach cone and sonic boom.

(Pictures taken from http://www.acs.psu.edu)

# What is Mass? A Short and Simple Explanation

Mass is such a fundamental property of matter that it is hard to define without drifting into philosophical realms. Newton’s Second Law provides a great way to understand mass from a physical point of view. The law states that force F (in N) is the product of mass m (in kg) and acceleration a (in m/s²):

F = m · a

So according to this, mass is a measure of an object’s resistance to a change in speed. If the mass is small, a small force is sufficient to produce a noticeable acceleration. However, much more force is necessary to produce the same acceleration for a massive object.

Another way of looking at mass is provided by Newton’s Law of Gravitation. Newton found that the attracting gravitational force between two objects is proportional to the product of their masses m and M:

F ~ m · M

So additionally to creating resistance to changes in state of motion, mass is also the source of gravitational attraction. It seems obvious that in both cases we are talking about the same quantity. But is this actually the case? Is the inertial mass, the mass responsible for opposing changes in velocity, really the same as the gravitational mass, that gives rise to gravity?

This question has led to heated debates among physicist for centuries. All experiments conducted so far, with ever increasing accuracy, have shown that indeed the inertial mass is identical to the gravitational mass. Today, almost all physicists have accepted this equivalence as reality.

The SI unit of mass is kilograms. Ever since 1889, one kilogram has been defined as the mass of the international prototype kilogram (IPK) that is stored in the International Bureau of Weights and Measures in Paris. However, during the 24th General Conference on Weights and Measures that took place in 2011, physicists have agreed to redefine this unit by connecting it to the Planck constant.

Other units that are commonly used for mass are grams (1/1000 of a kilogram), the pound (equal to about 0.45 kilograms) and the tonne (equal to 1000 kilograms). For atoms and molecules scientists use the atomic mass unit u. One u is equivalent to 1.66 · 10-27 kg, which is roughly the mass of a neutron or proton.

(This was an excerpt from Physics! In Quantities and Examples)